In the 1980s, van den Berg speculated that for all parallelepipeds the gap between the first two Dirichlet eigenvalues is bounded below by a constant. Yau subsequently formulated the fundamental gap conjecture:

For all convex domains in $\R^n$, the gap between the first two Dirichlet eigenvalues is bounded below by $\frac{3 \pi^2}{d^2}$, where $d^2$ is the diameter of the domain.

This talk concerns the spectral gap between Dirichlet eigenvalues of convex domains in $\R^n$, and in particular, the fundamental gap of simplices and triangles. I will discuss recent progress with Z. Lu on the fundamental gap conjecture for triangles and simplices, new connections between Neumann eigenvalues and Dirichlet gaps, and demonstrate a relationship between the fundamental gap and Bakry-Emery geometry. In conclusion, I will offer ideas and open problems.